and Heat Transfer between Stretching and Porous Surfaces Using Analytical Method

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Volume 2011, Article ID 258734,17pages doi:10.1155/2011/258734

Research Article

Investigation of Rotating MHD Viscous Flow

and Heat Transfer between Stretching and Porous Surfaces Using Analytical Method

M. Sheikholeslami,

¹

H. R. Ashorynejad,

¹

D. D. Ganji,

¹

and A. Kolahdooz

²

1Faculty of Mechanical Engineering, Babol University of Technology, Babol, Iran

2Faculty of Mechanical Engineering, Eslamic Azad University of Technology, Khomeninishahr Branch, Iran

Correspondence should be addressed to A. Kolahdooz,[email protected] Received 17 May 2011; Accepted 29 July 2011

Academic Editor: Ezzat G. Bakhoum

Copyrightq2011 M. Sheikholeslami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Hydromagnetic flow between two horizontal plates in a rotating system, where the lower plate is a stretching sheet and the upper is a porous solid plate, is analyzed. Heat transfer in an electrically conducting fluid bonded by two parallel plates is studied in the presence of viscous dissipation.

The equations of conservation of mass and momentum and energy are reduced to a nonlinear ordinary differential equations system. Homotopy perturbation method is used to get complete analytic solution for velocity and temperature profiles. Results show an acceptable agreement between this method results and numerical solution. Also the effects of different parameters are discussed through graphs.

1. Introduction

Flow of a viscose fluid over a stretching surface has important applications in polymer industries. For instance, a number of technical processes concerning polymers involve the cooling of continuous strips extruded from a die by drawing them through a quiescent fluid with controlled cooling system, and in the process of drawing, these strips are sometimes stretched.

Glass blowing, continuous casting of metals, and spinning of fibers also involve the flow over a stretching surface. In all these cases, the quality of the final product depends on the rate of heat transfer on the stretching surface.

Dutta et al. 1 studied the temperature field in the flow over a stretching surface subjected to uniform heat flux. Andersson et al. 2 investigated the unsteady two-dimensional non-Newtonian flow of a power-law fluid past a stretching surface.

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Bujurke et al. 3 and Dandapat and Gupta 4examined the temperature distribution in the steady boundary layers of a second-grade fluid near a stretching surface. P. S. Gupta and A. S. Gupta5investigated the heat and mass transfer on a stretching sheet with suction or blowing. Sakiadis6firstly studied the boundary layer flow over a stretched surface moving with constant velocity. Erickson et al.7extended the work of Sakiadis to include blowing or suction at the stretched sheet surface on a continuous moving surface with constant speed and investigated its eﬀects on the heat and mass transfer in the boundary layer.

In recent years, the eﬀect of magnetic field in diﬀerent engineering applications such as the cooling of reactors and many metallurgical processes involve the cooling of continuous tiles has been under more considerable attention. Several engineering processes, such as materials manufactured by extrusion processes and heat-treated materials traveling between a feed roll and a wind-up roll on convey belts possess the characteristics of a moving continuous surface, are just some examples of applications which involve the problem discussed above.

Chakrabarti and Gupta8studied the MHD flow of Newtonian fluids initially at rest, over a stretching sheet at a different uniform temperature. Vajravelu and Hadjinicolaou9 made analysis to flows and heat transfer characteristics in an electrically conducting fluid near an isothermal sheet. Heat transfer analysis of MHD fluid over a uniformly stretching sheet was investigated by Chakrabarti and Gupta10; In 1983, Borkakoti and Bharali11 studied the two-dimensional channel flow with heat transfer analysis of a hydromagnetic fluid where the lower plate was a stretching sheet. The flow between two rotating disks has important technical applications such as lubrication. Keeping this fact in mind, Vajravelu and Kumar 12studied the effect of rotation on the two-dimensional channel flow. They solved the governing equations analytically and numerically. Most of engineering problems, especially some of heat transfer equations, are nonlinear; therefore, some of them are solved using numerical solution, and some are solved using the different analytic method, such as perturbation method, homotopy perturbation method, and variational iteration method introduced by He13,14.

Perturbation techniques are based on the existence of small or large parameters, the so-called perturbation quantity. Unfortunately, many nonlinear problems in science and engineering do not contain those kinds of perturbation quantities. Therefore, many diﬀerent methods have recently introduced some ways to eliminate the small parameter. One of the semiexact methods which does not need small parameters is the homotopy perturbation method.

The homotopy perturbation method was proposed first by He in 1998 and was further developed and improved by He 15. The method yields a very rapid convergence of the solution series in the most of cases. The HPM proved its capability to solve a large class of nonlinear problems eﬃciently, accurately, and easily with approximations convergency very rapidly to solution. Usually, few iterations lead to high-accuracy solution. Recently, this method is employed for many researches in engineering sciences. He’s homotopy perturbation method is applied to obtain approximate analytical solutions for the motion of a spherical particle in a plane couette flow Jalaal et al.16. Then Jalaal et al.17showed the eﬀectiveness of HPM for unsteady motion of a spherical particle falling in a Newtonian fluid.

Ghotbi et al.18used HPM to approximate the solution of the ratio-dependent predator- prey system with constant eﬀort prey harvesting. Also homotopy perturbation method was used for solving nonlinear MHD Jeﬀery Hamel problem by Moghimi et al.19. Recently, Ganji et al. studied the steady-state flow of a Hagen-Poiseuille model in a circular pipe and entropy generation due to fluid friction and heat transfer using HPM20.

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B uw

uw

h

Th

Ω

x y z

v0 T0

Figure 1: Geometry of the problem.

In this study, the purpose is to solve nonlinear equations through the HPM. It can be seen that this method is strongly capable of solving a large class of coupled and nonlinear diﬀerential equations without tangible restriction of sensitivity to the degree of the nonlinear term.

2. Flow Analysis

2.1. Governing Equations

Consider the steady flow of an electrically conducting fluid between two horizontal parallel plates when the fluid and the plates rotate together around they-axis which is normal to the plates with an angular velocity.

A Cartesian coordinate system is considered as followes: thex-axis is along the plate, they-axis is perpendicular to it, and thez-axis is normal to thexyplaneseeFigure 1. The origin is located on the lower plate, and the plates are located aty0 andy h. The lower plate is being stretched by two equal and opposite forces, so that the position of the point 0,0,0remains unchanged. A uniform magnetic flux with densityB₀is acting alongy-axis about which the system is rotating. The upper plate is subjected to a constant flow injection with a velocityv0. The governing equations of motion in a rotating frame of reference are

∂u

∂x

∂v

∂y

∂w

∂z 0, 2.1

u∂u

∂x ν∂u

∂y 2Ωw−1 ρ

∂p^∗

∂x υ ∂²u

∂x²

∂²u

∂y²

−σB₀²

ρ u, 2.2

u∂v

∂y −1 ρ

∂p^∗

∂y υ ∂²v

∂x²

∂²v

∂y²

, 2.3

u∂w

∂x ν∂w

∂y −2 Ωw υ ∂²w

∂x²

∂²w

∂y²

−σB²₀

ρ w, 2.4

whereu,v, andwdenote the fluid velocity components along the x,y, andzdirections,υ is the kinematic coeﬃcient of viscosity,ρ is the fluid density, andp^∗ is the modified fluid pressure. The absence of∂p^∗/∂zin2.4implies that there is a net cross-flow along thez-axis.

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The boundary conditions are

uax, v0, w0 aty0,

u0, v−v0, w0 aty h. 2.5

The following nondimensional variables are introduced:

η y

h , uaxf η

, ν−ahf η

, waxg η

, 2.6

where a prime denotes diﬀerentiation with respect toη.

Substituting2.6in2.1–2.4, we have

− 1 ρh

∂p^∗

∂η a²x

f−ff−f R

M R

2Kr

R g

, 2.7

− 1 ρh

∂p^∗

∂η a²h

ff 1 Rf

, 2.8

g−R

fg−fg

2Krf−Mg0, 2.9

and the nondimensional quantities are defined, in whichRis the viscosity parameter,Mis the magnetic parameter, andKris the rotation parameter

R ah²

υ , M σB²₀h²

ρυ , Kr Ωh²

υ . 2.10

Equation2.7with the help of2.8can be written as follows:

f−R

f²−ff −2K²_rg−M²fA. 2.11

Diﬀerentiation of2.11with respect toηgives

f^iv−R

ff−ff

−2 Krg−Mf 0. 2.12

Therefore, the governing equations and boundary conditions for this case in nondimensional form are given by

f^iv−R

ff−ff

−2K_rg−Mf0, g−R

fg−fg

2Krf−Mg0,

2.13

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subject to the following boundary conditions:

f 0, f1, g 0 atη0, fλ, f0, g0 atη1,

λ v₀ ah.

2.14

3. Heat Transfer Analysis

3.1. Energy Equation

The energy equation for the present problem with viscous dissipation in nondimensional form is given by

u∂T

∂x v∂T

∂y w∂T

∂z k ρ cp

∂²T

∂x²

∂²T

∂y²

∂²T

∂z²

μϕ,

φ2 ∂u

∂x

₂ ∂v

∂y

₂ ∂w

∂z

₂ ∂v

∂x

∂u

∂y

₂ ∂w

∂y

∂v

∂z ₂

∂w

∂x

∂u

∂z 2

−2 3

∂u

∂x

∂v

∂y

∂w

∂z 2

.

3.1

With replacing nondimensional variables and using similarity solution method, by neglecting the last term of viscous dissipation in the energy equation, we have the following energy equation:

θ Pr

Rfθ Ec

4f² g² Ecx

f² g² 0, 3.2

subject to the boundary conditions

θ0 1, θ1 0, 3.3

where Pr μC_p/k is the Prandtl number, Ec a²h²/C_pθ0 −θ_h is the Eckert number, Ec_x a²x²/C_pθ0−θ_his the local Eckert number, and the nondimensional temperature is defined as

θ η

T−Th

T₀−T_h, 3.4

whereT₀andT_hare temperature at the lower and upper plates.

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f

0 0.2 0.4 0.6 0.8 1

η M=0 M=5

M=10 M=50 0

0.2 0.4

a

f^′

0 0.2 0.4 0.6 0.8 1

η M=0

M=5

M=10 M=50 0

0.2 0.4 0.6 0.8 1

b

g

0 0.2 0.4 0.6 0.8 1

η M=0 M=5

M=10 M=50 0

0.02 0.04 0.06

c

Figure 2: Velocity components profileaf,bf, andcgfor variableMatR 2,Kr 0.5, Pr 1, λ0.5, and EcEcx0.5.

4. Analysis of the Homotopy Perturbation Method

To illustrate the basic ideas of this method, we consider the following equation:

Au−fr 0 r∈Ω, 4.1

with the boundary condition of

B

u,∂u

∂n

0 r∈Γ, 4.2

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0 0.2 0.4 0.6 0.8 1 η

0 0.5 1 1.5 2

λ=0.5 λ=1

λ=1.5 λ=2 f

a

0 0.2 0.4 0.6 0.8 1

η 0

0.5 1 1.5 2

λ=0.5 λ=1

λ=1.5 λ=2 2.5

3

f^′

b

00

0.2 0.4 0.6 0.8 1

η λ=0.5 λ=1

λ=1.5 λ=2 0.05

0.1 0.15 0.2

g

c

Figure 3: Velocity components profileaf,bf, andcgfor variableλatR2,Kr0.5,M1, Pr1, and EcEcx0.5.

whereAis a general diﬀerential operator,Bis a boundary operator,fris a known analytical function, andΓis the boundary of the domainΩ.

A can be divided into two parts which are L and N, where L is linear and N is nonlinear. Equation4.1can therefore be rewritten as follows:

Lu Nu−fr 0 r ∈Ω. 4.3

Homotopy perturbation structure is shown as follows:

H ν, p

1−p

Lν−Lu0 p

Aν−fr

0, 4.4

where

ν r, p

:Ω×0,1−→R. 4.5

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Kr=0.5 Kr=2

Kr=4 Kr=6 g

0 0.2 0.4 0.6 0.8 1

η

a

0 0.1 0.2 0.3 0.4

R=5 R=10

R=15 R=20 g

0 0.2 0.4 0.6 0.8 1

η

b

Figure 4: Velocity components profilegfor variablesKrandRataR2,M1,λ0.5 andbKr4, M1,λ0.5, and EcEcx0.5.

In2.5,p ∈ 0,1is an embedding parameter, andu₀ is the first approximation that satisfies the boundary condition. We can assume that the solution of4.5can be written as a power series inp, as follows:

νν0 pν1 p²ν2 · · ·, 4.6

and the best approximation for solution is

ulim

p→1νν0 ν1 ν2 · · ·. 4.7

5. Implementation of the Method

According to the so-called homotopy perturbation methodHPM, we construct a homotopy.

Suppose the solution of4.1has the following form:

H f, p

1−p

f^iv −f₀^iv p

f^iv−R

ff− ff

−2 Krg−Mf 0, H

g, p

1−p

g −g₀ p

g −R

fg−fg

2K_rf−Mg 0, H

θ, p

1−p

θ −θ₀ p

θ Pr

R f θ Ec

4f² g² Ec_x

f² g² 0.

5.1

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0 0.2 0.4 0.6 0.8 1 η

0 0.2 0.4 0.6 0.8 1

θ

R=5 R=10

R=15 R=20 a

0 0.2 0.4 0.6 0.8 1

η 0

θ

0.3 0.6 0.9 1.2 1.5

Pr=0.7 Pr=7

Pr=10 Pr=40 b

0 0.2 0.4 0.6 0.8 1

η 0

0.5 1 1.5 2 2.5 3

λ=0.5 λ=1

λ=1.5 λ=2 θ

c

Figure 5: Temperature profileθfor variablesR, Pr andλatKr0.5,M1, and EcEcx0.5 anda λ0.5 and Pr1,bR2 andλ0.5, andcR2, Pr1.

We considerf, g, θas follows:

f η

f₀ η

pf₁ η

· · ·ⁿ

i0

pⁱf_i η

,

g η

g₀ η

pg₁ η

· · ·ⁿ

i0

pⁱg_i η

,

θ η

θ0

η pθ1

η

· · ·ⁿ

i0pⁱθi η

.

5.2

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with substitutingf, g, θfrom5.2into5.1and some simplification and rearranging based on powers ofp-terms, it can be obtained that

p⁰:

f₀^iv0, g₀0, θ₀0.

And boundary conditions are

f0 0, f0 1, f1 λ, f1 0, g0 1, g1 0,

θ0 1, θ1 0,

5.3

p¹:

−g₀ −0.5f₀ 0.5f₀f₀−0.5f₀f₀ f₁^iv0,

−0.5f₀g₀−0.5g₀ g₁ 0.5f₀g₀ f₀ 0, 0.25

f₀2 0.25

g₀2 0.25

g₀2 θ₁ 0.25f₀θ₁

f₀20.

f0 0, f0 0, f1 0, f1 0, g0 0, g1 0,

θ0 0, θ1 0,

5.4

p²:

0.5f0f₁−g₀ 0.5f₁f₀ 0.5f₁f₀−0.5f₁ f₂^iv0, 0.5f₁g₀ 0.5f₀g₁ f₀−0.5g₁−0.5f₀g₁ g₂−0.5f₁g₀0, 0.25f₀θ₁ 0.25f₁θ₀ 0.25

f₀2 θ₂ 2f₀f₁ 0.5f₀f₁ 0.5g₀g₁ 0.5g₀g₁ 0.

f0 0, f0 0, f1 0, f1 0, g0 0, g1 0,

θ0 0, θ1 0.

5.5

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By solving5.3–5.5with boundary conditions forR 0.5,Kr 0.5,M0.5, Pr 0.5,λ0.5, and EcEc_x0.5, it can be obtained that

f₀ η

−0.5η² η, g₀

η 0, θ0

η

−η 1, f₁

η

−0.0007η⁶ 0.0083η⁵−0.0416η⁴ 0.0611η³−0.0270η², g1

η

0.1667η³−0.5η² 0.3333x, θ₁

η

−0.0937η⁴ 0.3750η³−0.6250η² 0.3437η, f2

η

0.0002η⁸ 0.0010η⁷−0.0116η⁶−0.0046η⁵ 0.0116η⁴−0.0081η³ 0.0016η², g₂

η

0.0041η⁵−0.0222η⁴ 0.0368η³−0.0188η, θ2

η

−0.0001η⁸ 0.0011η⁷−0.0067η⁶ 0.0180η⁵−0.0313η⁴ 0.0342η³

−0.0135η²−0.0016η.

5.6

The solution of this equation, whenp → 1, will be as follows:

f η

f₀ η

f₁ η

· · · f₇ η

f₈ η

, g

η g0

η g1

η

· · · g7

η g8

η , θ

η θ₀

η θ₁

η

· · · θ₇ η

θ₈ η

.

5.7

where forR 0.5,Kr 0.5,M 0.5, Pr 0.5,λ 0.5, and Ec Ecx 0.5, the following functions are obtained:

f η

−0.0001η⁸ 0.0007η⁷−0.0012η⁶ 0.0033η⁵−0.0306η⁴ 0.0537η³−0.5256η² η, g

η

−0.0001η⁸ 0.0002η⁷ 0.0009η⁶ 0.0019η⁵−0.0203η⁴ 0.2015η³−0.5η² 0.3158η, θ

η

0.0001η⁹−0.0005η⁸ 0.0015η⁷−0.0093η⁶ 0.0293η⁵−0.1481η⁴ 0.4322η³

−0.6506η²−0.6546η 1.

5.8

6. Results and Discussion

The objective of the present study was to apply homotopy perturbation method to obtain an explicit analytic solution of rotating MHD flow and heat transfer of viscous fluid over stretching and porous surfaceFigure 1. As can be seen inTable 1, homotopy perturbation method is converged in step 8, and error has been minimized. There is an acceptable

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Table 1:θηvalues in diﬀerent steps of HPM solution atR0.5,Kr0.5,M0.5, Pr0.5,λ0.5, and EcEcx0.5.

η NM n2 % Error n4 % Error n6 % Error n8 % Error

0 1 1 0 1 0 1 0 1 0

0.1 0.928387 0.928223 0.017678 0.928444 0.006163 0.928449 0.006698635 0.928449 0.00670429 0.2 0.846138 0.84596 0.021041 0.846272 0.01581 0.84628 0.016736093 0.84628 0.016746103 0.3 0.755385 0.755242 0.018984 0.755583 0.026195 0.755592 0.027417617 0.755592 0.027431147 0.4 0.657944 0.657829 0.017493 0.658177 0.035331 0.658186 0.036816201 0.658187 0.036832934 0.5 0.555344 0.555229 0.020556 0.555575 0.041731 0.555585 0.043510198 0.555585 0.043530456 0.6 0.44884 0.448702 0.030858 0.449039 0.044167 0.449048 0.046342095 0.449048 0.046367093 0.7 0.339433 0.339262 0.050375 0.339574 0.041441 0.339583 0.044186822 0.339583 0.044218728 0.8 0.227869 0.227685 0.080907 0.227943 0.03217 0.227951 0.035722981 0.227951 0.035765242 0.9 0.114645 0.114502 0.124586 0.114662 0.014558 0.114667 0.019207699 0.114667 0.019265704

1 0 0 0 0 0 0 0 0 0

Table 2: Comparison between numerical results and HPM solution forf,g,θatR0.5,Kr0.5,M0.5, Pr0.5,λ0.5, and EcEcx0.5.

η f g θ

NM HPM % Error NM HPM % Error NM HPM % Error

0 0 0 0 0 0 0 1 1 0

0.1 0.094742 0.094794 0.054323 0.026758 0.026778 0.073964 0.928387 0.928449 0.006224175 0.2 0.179187 0.179354 0.093254 0.044707 0.044738 0.069201 0.846138 0.84628 0.014169522 0.3 0.253603 0.253898 0.116124 0.054988 0.055018 0.054436 0.755385 0.755592 0.020721081 0.4 0.318182 0.318574 0.123244 0.058698 0.058716 0.030566 0.657944 0.658187 0.024234016 0.5 0.373035 0.373467 0.115934 0.056892 0.056891 0.000929 0.555344 0.555585 0.02417436 0.6 0.418197 0.418601 0.096615 0.050587 0.050568 0.037868 0.44884 0.449048 0.020811422 0.7 0.453627 0.45394 0.068972 0.040777 0.040746 0.077163 0.339433 0.339583 0.015009305 0.8 0.479208 0.479391 0.038258 0.028434 0.028402 0.114496 0.227869 0.227951 0.008149803 0.9 0.494751 0.49481 0.011815 0.014521 0.0145 0.143769 0.114645 0.114667 0.002208717

1 0.5 0.5 0 0 0 0 0 0 0

agreement between the results of numerical solution obtained by four-order Rung-kutte method and diﬀerential transformation method as shown in Tables2and3. In those tables, error is introduced as follows:

%Error

f η

NM−f η

HPM

f η

NM

×100. 6.1

Figure 2shows the magnetic field eﬀect on nondimensional velocity componentf,f, andg. The decrease off curve is observed by applying higher magnetic field intensity, and fvalues increase near stretching sheet and decrease under porous sheet, while at the middle point, these values are constant.

At low Reynolds numbers, the velocity profile exhibits center line symmetry indicating a Poiseuille flow for non-Newtonian fluids. At higher Reynolds numbers, the maximum velocity point is shifted to the streiching wall where shear stress becomes larger as the Reynolds number grows.

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Table 3: Comparison between numerical results and HPM solution forf,g,θatR0.5,Kr0.5,M0, Pr0.5,λ0.5, and EcEcx0.5.

η f g θ

NM HPM % Error NM HPM % Error NM HPM % Error

0 0 0 0 0 0 0 1 1 0

0.1 0.094908 0.094961 0.055277 0.027788 0.02781 0.076047 0.9286 0.928667 0.007196834 0.2 0.179709 0.179881 0.095531 0.046609 0.046643 0.071459 0.846601 0.846752 0.017836439 0.3 0.254498 0.254803 0.119875 0.057507 0.057539 0.056888 0.75604 0.756261 0.029298263 0.4 0.319345 0.319755 0.128301 0.061531 0.061552 0.033084 0.658692 0.658953 0.039612508 0.5 0.374292 0.374748 0.121774 0.059736 0.059736 0.001376 0.556078 0.556341 0.047236899 0.6 0.419351 0.419781 0.102421 0.05317 0.053151 0.036168 0.449468 0.449697 0.050848494 0.7 0.454508 0.454843 0.073797 0.04288 0.042847 0.076493 0.339887 0.340054 0.049140518 0.8 0.479718 0.479916 0.041304 0.029902 0.029868 0.11517 0.228122 0.228215 0.040617504 0.9 0.494912 0.494976 0.012863 0.015268 0.015246 0.14576 0.114722 0.114749 0.023395295

1 0.5 0.5 0 0 0 0 0 0 0

0 0.5 1 1.5

Ec=0.5 Ec=2

Ec=4 Ec=6

0 0.2 0.4 0.6 0.8 1

η θ

a

0 0.5 1 1.5

Ecx=0.5 Ecx=2

Ecx=4 Ecx=6

0 0.2 0.4 0.6 0.8 1

η θ

b

Figure 6: Temperature profile for variables Ec and Ecx atR2,Kr0.5,M1, Pr1, andλ0.5 and aEcx0.5,bEc0.5.

Blowing velocity parameter λ has a noticable eﬀect of nondimensional velocity component as shown inFigure 3, which by increasingλprofile offandfbecomes nonlinear, and the maximum amount of f and f increases, and velocity component in x direction increases severely.

Also it shows that increasing the blowing velocity parameter leads togincrease, which shows that blowing velocity parameter and magnetic field eﬀects ongare in opposite.

Figure 4 shows that by increasing rotating parameter Kr, values of transverse velocity componentgbetween two sheets increase, and the location of maximum amount ofg approaches stretching sheet. Coriolis force has inverse eﬀect ong in comparison with Lorenz force which means that with increasing the rotating parameter transverse Kr, velocity component between two plates increases as shown inFigure 4. And it shows that

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0 1 2 3 4

−4

−2 0 2 4

Kr

f^′′(1) f^′′(0)

a

0.5 1 1.5 2 2.5

−10

−5 0 5 10

λ

f^′′(1) f^′′(0)

b

5 10 15 20

−4

−2 0 2 4 6

R

f^′′(1) f^′′(0)

c

0 10 20 30 40 50

−8

−6

−4

−2 0 2 4

M

f^′′(1) f^′′(0)

d

Figure 7: The eﬀect of active parameters on skin friction at EcEcx 0.5 andaR0.5,M0.5, and λ1,bR0.5,Kr0.5, andM0.5,cKr0.5,M0.5, andλ1,dR0.5,Kr0.5, andλ1.

the viscosity parameter R aﬀects g profile similarly to magnetic field; however, with less changes in intensity, also with increasing R, the location of maximum amount of g approaching stretching sheet, hat indicates decreasing of boundary layer thickness near stretching plate.

As can be seen inFigure 5, increasing viscosity parameter leads to increasing the curve of temperature profileθand the decreasing ofθvalues, and it can be shown that increasing Pr in presence of viscous dissipation leads to increasing temperature between two plates.

Also increasing temperature between two plates observed, which is caused by increasing this eﬀect, is more sensible near the stretching plate.

The eﬀects of viscous dissipation for which the Eckert numberEcand the local Eckert numberEcxare responsible are shown inFigure 6. It is obvious from the graphs that by

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5 10 15 20 25 0

0.4 0.8 1.2 1.6 2

θ′(0)

R a

0 1 2 3 4

1.5 2 2.5 3 3.5 4

θ′(0)

Kr

b

0.5 1 1.5 2 2.5

0 1.5 3 4.5 6 7.5 9

θ′(0)

λ c

10 20 30

0 30 60 90 120

θ′(0)

Pr d

Figure 8: The eﬀect of active parameters on Nusselt numberNuθ0.aKr 0.5, Pr0.5,λ1.5 bR0.5, Pr0.5,λ 1.5cR0.5,Kr 0.5,λ 1.5dR 0.5,Kr 0.5, Pr0.5 andM0.5, EcEcx0.5.

increasing Ec and Ecx, the temperature near the stretching wall increases. This is due to the fact that heat energy is stored in the fluid due to the frictional heating.

InFigure 7, a coefficient of skin friction in stretching platef0and porous plate f1is discussed with hanging effective parameters. In stretching plate, with the increase of viscosity parameter, skin friction increases, and increasing rotating parameter leads to a similar effect on skin friction. Applying higher magnetic field intensity leads to skin friction reduction. Increasing blowing velocity parameter leads to skin friction increasing. For porous plate, with the increase of R, skin friction decreases, while with M and Kr increase, the reduction in skin friction observed; is also withλreduction, skin friction increases.

A coeﬃcient of Nusselt number θ0 consulted changes of eﬀective parameters.

Increasing theMorRleads to Nu decreasing, while by increasingKr, Pr, andλ, the Nusselt number increases, as shown inFigure 8.

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7. Conclusion

In this paper, hydromagnetic flow problem between two horizontal plates in a rotating system, where the lower plat is a stretching sheet and the upper is a porous solid plate, has been solved via a sort of analytical method, homotopy perturbation method. Also this problem is solved by a numerical methodthe Runge-Kutta method of order 4, and some of the conclusions were summarized as follows:

ahomotopy perturbation method is a powerful approach for solving nonlinear diﬀerential equation such as the discussed problem, and it can be observed that there is a good agreement between the present and numerical results;

bpresence of magnetic field leads to creating a Lorentz force which causes transverse velocity component reduction between two plates although this force does not have a noticeable eﬀect on temperature profile;

cincreasing Pr in presence of viscous dissipation leads to temperature increasing between two plates, while in absence of viscous dissipation, the changes are inverse;

dincreasing temperature between two plates is due to increasing viscosity parameter or increasing viscous dissipation, whose eﬀect is more sensible near stretching plate;

eincreasing magnetic parameter or viscosity parameter leads to decreasing Nu, while with increasing the rotation parameter, blowing velocity parameter, and Pr, the Nusselt number increases.

Nomenclature

B0: Magnetic fieldwb·m⁻² M: Magnetic parameter Ci: Constant function R: Viscosity parameter K_r: Rotation parameter p^∗: Modified fluidpressure Pr: Prandtl number

Ec: Eckert number v₀: Injection velocity

Umax: Maximum value of velocity

u,v,w: Velocity components alongx,y, andzaxes, respectively.

Greek Symbols

υ: Kinematic viscosity α: Angle of the channel θ: Any angle

η: Dimensionless angle ρ: Fluid density.

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References

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3 N. M. Bujurke, S. N. Biradar, and P. S. Hiremath, “Second-order fluid flow past a stretching sheet with heat transfer,” Zeitschrift f ¨ur Angewandte Mathematik und Physik, vol. 38, no. 4, pp. 653–657, 1987.

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International Journal of Non-Linear Mechanics, vol. 24, no. 3, pp. 215–219, 1989.

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Journal of Chemical and Engineering Data, vol. 55, pp. 744–745, 1977.

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26–28, 1961.

7 L. E. Erickson, L. T. Fan, and V. G. Fox, “Heat and mass transfer on a moving continuous flat plate with suction or injection,” Industrial and Engineering Chemistry Fundamentals, vol. 5, no. 1, pp. 19–25, 1966.

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Quarterly of Applied Mathematics, vol. 37, no. 1, pp. 73–78, 1979.

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13-12, pp. 1237–1244, 1997.

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461–467, 1983.

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1, pp. 13–24, 2004.

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2267–2270, 2011.

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8, pp. 2213–2216, 2011.

20 D. D. Ganji, H. R. Ashory Nezhad, and A. Hasanpour, “Eﬀect of variable viscosity and viscous dissipation on the Hagen-Poiseuille flow and entropy generation,” Numerical Methods for Partial Diﬀerential Equations, vol. 27, no. 3, pp. 529–540, 2011.

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and Heat Transfer between Stretching and Porous Surfaces Using Analytical Method

Research Article

Investigation of Rotating MHD Viscous Flow

and Heat Transfer between Stretching and Porous Surfaces Using Analytical Method

M. Sheikholeslami,

H. R. Ashorynejad,

D. D. Ganji,

and A. Kolahdooz

1. Introduction

2. Flow Analysis

3. Heat Transfer Analysis

4. Analysis of the Homotopy Perturbation Method

5. Implementation of the Method

6. Results and Discussion

7. Conclusion

Nomenclature

Greek Symbols

References

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